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algebraic data types and highlight the relationship between the implementation and the reference mathematical equations [[2]](#2).\n\nSGP4 can be called from JavaScript or Python via WebAssembly wrappers. See https://github.com/wasmerio/sgp4 to install and use SGP4 as a WAPM package.\n\nThe numerical predictions are almost identical to Celestrak's. The observed differences (less than 2 × 10⁻⁷ km for the position and 10⁻⁹ km.s⁻¹ for the velocity three and a half years after the epoch) are well below the accuracy of the algorithm.\n\nWe drew inspiration from the incomplete https://github.com/natronics/rust-sgp4 to write mathematical expressions using UTF-8 characters.\n\n- [Documentation](#documentation)\n- [Environments without std or alloc](#environments-without-std-or-alloc)\n- [Benchmark](#benchmark)\n- [Variables and mathematical expressions](#variables-and-mathematical-expressions)\n - [Variables](#variables)\n - [Initialization variables](#initialization-variables)\n - [Propagation variables](#propagation-variables)\n - [Third-body initialization variables](#third-body-initialization-variables)\n - [Third-body propagation variables](#third-body-propagation-variables)\n - [Mathematical expressions](#mathematical-expressions)\n - [UT1 to Julian conversion](#ut1-to-julian-conversion)\n - [Common initialization](#common-initialization)\n - [Near earth initialization](#near-earth-initialization)\n - [High altitude near earth initialization](#high-altitude-near-earth-initialization)\n - [Elliptic high altitude near earth initialization](#elliptic-high-altitude-near-earth-initialization)\n - [Deep space initialization](#deep-space-initialization)\n - [Third body perturbations](#third-body-perturbations)\n - [Resonant deep space initialization](#resonant-deep-space-initialization)\n - [Geosynchronous deep space initialization](#geosynchronous-deep-space-initialization)\n - [Molniya deep space initialization](#molniya-deep-space-initialization)\n - [Common propagation](#common-propagation)\n - [Near earth propagation](#near-earth-propagation)\n - [High altitude near earth propagation](#high-altitude-near-earth-propagation)\n - [Deep space propagation](#deep-space-propagation)\n - [Third body propagation](#third-body-propagation)\n - [Resonant deep space propagation](#resonant-deep-space-propagation)\n - [Lyddane deep space propagation](#lyddane-deep-space-propagation)\n- [References](#references)\n\n## Documentation\n\nThe code documentation is hosted at [https://docs.rs/sgp4/latest/sgp4/](https://docs.rs/sgp4/latest/sgp4/).\n\nExamples can be found in this repository's _examples_ directory:\n\n- _examples/celestrak.rs_ retrieves the most recent Galileo OMMs from Celestrak and propagates them\n- _examples/omm.rs_ parses and propagates a JSON-encoded OMM\n- _examples/space-track.rs_ retrieves the 20 most recent launches OMMs from Space-Track and propagates them\n- _examples/tle.rs_ parses and propagates a TLE\n- _examples/tle_afspc.rs_ parses and propagates a TLE using the AFSPC compatibility mode\n- _examples/advanced.rs_ leverages the advanced API to (marginally) accelerate the propagation of deep space resonant satellites\n\nTo run an example (here _examples/celestrak.rs_), use:\n\n```sh\ncargo run --example celestrak\n```\n\nTo run the Space-Track example, you must first assign your Space-Track.org credentials to the fields `identity` and `password` (see lines 3 and 4 in _examples/space-track.rs_).\n\n## Environments without std or alloc\n\nThis crate supports `no_std` environments. TLE parsing and SGP4 propagation do not require `alloc` either. We use [num-traits](https://docs.rs/num-traits/latest/num_traits/) with [libm](https://docs.rs/libm/latest/libm/) for floating point functions when `std` is not available.\n\nSee https://github.com/neuromorphicsystems/sgp4-no-std for a minimal no-std example that runs on Docker Linux x86_64.\n\nAll serde-related features, such as OMM parsing, require `alloc`.\n\n## Benchmark\n\nThe benchmark code is available at https://github.com/neuromorphicsystems/sgp4-benchmark. It compares two SGP4 implementations in different configurations:\n\n- `cpp`: the Celestrak implementation [[1]](#1) in improved mode\n- `cpp-afspc`: the Celestrak implementation [[1]](#1) in AFSPC compatibility mode\n- `cpp-fastmath`: the Celestrak implementation [[1]](#1) in improved mode with the `fast-math` compiler flag\n- `cpp-afspc-fastmath`: the Celestrak implementation [[1]](#1) in AFSPC compatibility mode with the `fast-math` compiler flag\n- `rust`: our Rust implementation in default mode\n- `rust-afspc`: our Rust implementation in AFSPC compatibility mode\n\nThis benchmark must not be confused with the code in this repository's _bench_ directory. The latter considers only a small subset of the Celestrak catalogue (the tests recommended in [[1]](#1)) and does not measure the original C++ implementation.\n\nThe present results were obtained using a machine with the following configuration:\n\n- **CPU** - Intel Core i7-8700 @ 3.20GHz\n- **RAM** - Kingston DDR4 @ 2.667 GHz\n- **OS** - Ubuntu 16.04\n- **Compilers** - Rust 1.44.1 and gcc 9.3.0\n\nAccuracy measures the maximum propagation error of each implementation with respect to the reference implementation (`cpp-afspc`) over the full Celestrak catalogue (1 minute timestep over 24 hours).\n\n| implementation | maximum position error | maximum speed error |\n| -------------------- | ---------------------- | ------------------- |\n| `cpp-afspc` | (reference) | (reference) |\n| `cpp` | 1.05 km | 1.30 × 10⁻³ km.s⁻¹ |\n| `cpp-fastmath` | 1.05 km | 1.30 × 10⁻³ km.s⁻¹ |\n| `cpp-afspc-fastmath` | 4.21 × 10⁻⁸ km | 7.51 × 10⁻¹² km.s⁻¹ |\n| `rust` | 1.05 km | 1.30 × 10⁻³ km.s⁻¹ |\n| `rust-afspc` | 4.19 × 10⁻⁸ km | 7.46 × 10⁻¹² km.s⁻¹ |\n\nThe Rust and C++ fast-math errors have the same order of magnitude. In both cases, they can be attributed to mathematically identical expressions implemented with different floating-point operations.\n\nSpeed measures the time it takes to propagate every satellite in the Celestrak catalogue (1 minute timestep over 24 hours) using a single thread. 100 values are sampled per implementation.\n\n| implementation | minimum | Q1 | median | Q3 | maximum | relative difference |\n| -------------------- | ------- | ------ | ------ | ------ | ------- | ------------------- |\n| `cpp-afspc` | 8.95 s | 9.02 s | 9.03 s | 9.06 s | 9.18 s | (reference) |\n| `cpp` | 8.95 s | 9.01 s | 9.04 s | 9.06 s | 9.25 s | + 0 % |\n| `cpp-fastmath` | 7.67 s | 7.74 s | 7.77 s | 7.79 s | 7.90 s | - 14 % |\n| `cpp-afspc-fastmath` | 7.70 s | 7.74 s | 7.76 s | 7.79 s | 7.86 s | - 14 % |\n| `rust` | 8.36 s | 8.41 s | 8.43 s | 8.45 s | 8.53 s | - 7 % |\n| `rust-afspc` | 8.36 s | 8.41 s | 8.43 s | 8.46 s | 8.59 s | - 7 % |\n\nRust fast-math support is a work in progress (see https://github.com/rust-lang/rust/issues/21690). Similarly to C++, it should have a very small impact on accuracy while providing a substantial speed gain.\n\n## Variables and mathematical expressions\n\n### Variables\n\nEach variable is used to store the result of one and only one expression. Most variables are immutable, with the exception of the variable `(E + ω)ᵢ` used to solve Kepler's equation and the state variables `tᵢ`, `nᵢ` and `λᵢ` used to integrate the resonance effects of Earth gravity.\n\nThe following tables list the variables used in the code and their associated mathematical symbol. Where possible, we used symbols from [[2]](#2). Partial expressions without a name in [[2]](#2) follow the convention `kₙ, n ∈ ℕ` if they are shared between initialization and propagation, and `pₙ, n ∈ ℕ` if they are local to initialization or propagation.\n\n1. [Initialization variables](#initialization-variables)\n2. [Propagation variables](#propagation-variables)\n3. [Third-body initialization variables](#third-body-initialization-variables)\n4. [Third-body propagation variables](#third-body-propagation-variables)\n\n---\n\n#### Initialization variables\n\nThe following variables depend solely on epoch elements.\n\n| variable | symbol | description |\n| :-------------------------------- | :------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n| `Elements::datetime.year()` | `yᵤ` | Gregorian calendar year |\n| `Elements::datetime.month()` | `mᵤ` | Gregorian calendar month in the range `[1, 12]` |\n| `Elements::datetime.day()` | `dᵤ` | Gregorian calendar day in the range `[1, 31]` |\n| `Elements::datetime.hour()` | `hᵤ` | Hours since midnight in the range `[0, 23]` |\n| `Elements::datetime.minute()` | `minᵤ` | Minutes since the hour in the range `[0, 59]` |\n| `Elements::datetime.second()` | `sᵤ` | Seconds since the minute in the range `[0, 59]` |\n| `Elements::datetime.nanosecond()` | `nsᵤ` | Nanoseconds since the second in the range `[0, 10⁹[` |\n| `epoch` | `y₂₀₀₀` | Julian years since UTC 1 January 2000 12h00 (J2000) |\n| `d1900` | `d₁₉₀₀` | Julian days since UTC 1 January 1900 12h00 (J1900) |\n| `d1970` | `d₁₉₇₀` | Julian days since UTC 1 January 1970 12h00 (J1970) |\n| `c2000` | `c₂₀₀₀` | Julian centuries since UTC 1 January 2000 12h00 (J2000) |\n| `geopotential.ae` | `aₑ` | equatorial radius of the earth in km |\n| `geopotential.ke` | `kₑ` | square root of the earth's gravitational parameter in earth radii³ min⁻² |\n| `geopotential.j2` | `J₂` | un-normalised second zonal harmonic |\n| `geopotential.j3` | `J₃` | un-normalised third zonal harmonic |\n| `geopotential.j4` | `J₄` | un-normalised fourth zonal harmonic |\n| `kozai_mean_motion` | `n₀` | mean number of orbits per day (Kozai convention) at epoch in rad.min⁻¹ |\n| `a1` | `a₁` | semi-major axis at epoch (Kozai convention) |\n| `p0` | `p₀` | partial expression of `𝛿₀` and `𝛿₁` |\n| `d1` | `𝛿₁` | used in the Kozai to Brouwer conversion |\n| `d0` | `𝛿₀` | used in the Kozai to Brouwer conversion |\n| `B*` | `B*` | radiation pressure coefficient in earth radii⁻¹ |\n| `orbit_0.inclination` | `I₀` | angle between the equator and the orbit plane at epoch in rad |\n| `orbit_0.right_ascension` | `Ω₀` | angle between vernal equinox and the point where the orbit crosses the equatorial plane at epoch in rad |\n| `orbit_0.eccentricity` | `e₀` | shape of the orbit at epoch |\n| `orbit_0.argument_of_perigee` | `ω₀` | angle between the ascending node and the orbit's point of closest approach to the earth at epoch in rad |\n| `orbit_0.mean_anomaly` | `M₀` | angle of the satellite location measured from perigee at epoch in rad |\n| `orbit_0.mean_motion` | `n₀\"` | mean number of orbits per day (Brouwer convention) at epoch in rad.min⁻¹ |\n| `p1` | `p₁` | cosine of the inclination at epoch used in multiple expressions during initialization (`θ` in [[2]](#2), renamed to avoid confusion with the sidereal time) |\n| `p2` | `p₂` | partial expression of multiple initialization expressions |\n| `a0` | `a₀\"` | semi-major axis at epoch (Brouwer convention) |\n| `p3` | `p₃` | perigee in earth radii |\n| `p4` | `p₄` | height of perigee in km |\n| `p5` | `p₅` | partial expression of `s` |\n| `s` | `s` | altitude parameter of the atmospheric drag expression |\n| `p6` | `p₆` | partial expression of the atmospheric drag |\n| `xi` | `ξ` | partial expression of multiple initialization expressions |\n| `p7` | `p₇` | partial expression of multiple initialization expressions |\n| `eta` | `η` | partial expression of multiple initialization expressions and of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation |\n| `p8` | `p₈` | partial expression of multiple initialization expressions |\n| `p9` | `p₉` | partial expression of multiple initialization expressions |\n| `c1` | `C₁` | partial expression of multiple initialization and propagation expressions |\n| `p10` | `p₁₀` | partial expression of multiple initialization expressions |\n| `b0` | `β₀` | partial expression of multiple initialization expressions |\n| `p11` | `p₁₁` | partial expression of multiple initialization expressions |\n| `p12` | `p₁₂` | partial expression of multiple initialization expressions |\n| `p13` | `p₁₃` | partial expression of multiple initialization expressions |\n| `p14` | `p₁₄` | partial expression of multiple initialization expressions |\n| `p15` | `p₁₅` | partial expression of multiple initialization expressions |\n| `k14` | `k₁₄` | first order coefficient of the argument of perigee before adding solar and lunar perturbations |\n| `c4` | `C₄` | partial expression of multiple initialization and propagation expressions (differs from the `C₄` expression in [[2]](#2) by a factor B\\*) |\n| `right_ascension_dot` | `Ω̇` | first order coefficient of the right ascension |\n| `argument_of_perigee_dot` | `ω̇` | first order coefficient of the argument of perigee |\n| `mean_anomaly_dot` | `Ṁ` | first order coefficient of the mean anomaly |\n| `k0` | `k₀` | second order coefficient of the right ascension before adding perturbations |\n| `k1` | `k₁` | partial expression of the second order coefficient of the mean anomaly |\n| `k2` | `k₂` | partial expression of `aᵧₙ` in near earth propagation |\n| `k3` | `k₃` | partial expression of `rₖ`, `ṙₖ` and `rḟₖ` in near earth propagation |\n| `k4` | `k₄` | partial expression of `uₖ` in near earth propagation |\n| `k5` | `k₅` | partial expression of the initial Kepler variable `p₃₈` in near earth propagation |\n| `k6` | `k₆` | partial expression of multiple initialization expressions and of `rₖ` and `rḟₖ` in near earth propagation |\n| `d2` | `D₂` | partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation |\n| `p16` | `p₁₆` | partial expression of multiple near earth initialization expressions |\n| `d3` | `D₃` | partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation |\n| `d4` | `D₄` | partial expression of multiple near earth initialization expressions and of the semi-major axis in near earth propagation |\n| `c5` | `C₅` | partial expression of multiple initialization and propagation expressions (differs from the `C₅` expression in [[2]](#2) by a factor B\\*) |\n| `k7` | `k₇` | sine of the mean anomaly at epoch |\n| `k8` | `k₈` | partial expression of the mean anomaly third order coefficient in high altitude near earth propagation |\n| `k9` | `k₉` | partial expression of the mean anomaly fourth order coefficient in high altitude near earth propagation |\n| `k10` | `k₁₀` | partial expression of the mean anomaly fifth order coefficient in high altitude near earth propagation |\n| `k11` | `k₁₁` | partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation |\n| `k12` | `k₁₂` | partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation |\n| `k13` | `k₁₃` | partial expression of the argument of perigee and mean anomaly in eccentric high altitude near earth propagation |\n| `lunar_right_ascension_epsilon` | `Ωₗₑ` | lunar right ascension of the ascending node |\n| `lunar_right_ascension_sine` | `sin Ωₗ` | sine of the lunar right ascension of the ascending node referred to the equator |\n| `lunar_right_ascension_cosine` | `cos Ωₗ` | cosine of the lunar right ascension of the ascending node referred to the equator |\n| `lunar_argument_of_perigee` | `ωₗ` | lunar argument of perigee |\n| `sidereal_time_0` | `θ₀` | Greenwich sidereal time at epoch |\n| `lambda_0` | `λ₀` | Earth gravity resonance variable at epoch |\n| `lambda_dot_0` | `λ̇₀` | time derivative of the Earth gravity resonance variable at epoch |\n| `p17` | `p₁₇` | partial expression of `𝛿ᵣ₁`, `𝛿ᵣ₂` and `𝛿ᵣ₃` |\n| `dr1` | `𝛿ᵣ₁` | first Earth gravity resonance coefficient for geosynchronous orbits (`𝛿₁` in [[2]](#2), renamed to avoid confusion with `𝛿₁` used in the Kozai to Brouwer conversion) |\n| `dr2` | `𝛿ᵣ₂` | second Earth gravity resonance coefficient for geosynchronous orbits (`𝛿₂` in [[2]](#2), renamed to match `𝛿ᵣ₁`) |\n| `dr3` | `𝛿ᵣ₃` | third Earth gravity resonance coefficient for geosynchronous orbits (`𝛿₃` in [[2]](#2), renamed to match `𝛿ᵣ₁`) |\n| `p18` | `p₁₈` | partial expression of `D₂₂₀₋₁` and `D₂₂₁₁` |\n| `p19` | `p₁₉` | partial expression of `D₃₂₁₀` and `D₃₂₂₂` |\n| `p20` | `p₂₀` | partial expression of `D₄₄₁₀` and `D₄₄₂₂` |\n| `p21` | `p₂₁` | partial expression of `D₅₂₂₀`, `D₅₂₃₂`, `D₅₄₂₁` and `D₅₄₃₃` |\n| `f220` | `F₂₂₀` | partial expression of `D₂₂₀₋₁` and `D₄₄₁₀` |\n| `g211` | `G₂₁₁` | partial expression of `D₂₂₁₁` |\n| `g310` | `G₃₁₀` | partial expression of `D₃₂₁₀` |\n| `g322` | `G₃₂₂` | partial expression of `D₃₂₂₂` |\n| `g410` | `G₄₁₀` | partial expession of `D₄₄₁₀` |\n| `g422` | `G₄₂₂` | partial expession of `D₄₄₂₂` |\n| `g520` | `G₅₂₀` | partial expression of `D₅₂₂₀` |\n| `g532` | `G₅₃₂` | partial expression of `D₅₂₃₂` |\n| `g521` | `G₅₂₁` | partial expression of `D₅₄₂₁` |\n| `g533` | `G₅₃₃` | partial expression of `D₅₄₃₃` |\n| `d220₋1` | `D₂₂₀₋₁` | gravity resonance coefficient for Molniya orbits (the `Dₗₘₚₖ` expression in [[2]](#2) is missing a factor `l - 2p + k` from the original equation in [[4]](#4) with `k = -1` instead of `1`) |\n| `d2211` | `D₂₂₁₁` | gravity resonance coefficient for Molniya orbits (the `Dₗₘₚₖ` expression in [[2]](#2) is missing a factor `l - 2p + k` from the original equation in [[4]](#4)) |\n| `d3210` | `D₃₂₁₀` | see `D₂₂₁₁` |\n| `d3222` | `D₃₂₂₂` | see `D₂₂₁₁` |\n| `d4410` | `D₄₄₁₀` | see `D₂₂₁₁` |\n| `d4422` | `D₄₄₂₂` | see `D₂₂₁₁` |\n| `d5220` | `D₅₂₂₀` | see `D₂₂₁₁` |\n| `d5232` | `D₅₂₃₂` | see `D₂₂₁₁` |\n| `d5421` | `D₅₄₂₁` | see `D₂₂₁₁` |\n| `d5433` | `D₅₄₃₃` | see `D₂₂₁₁` |\n\n#### Propagation variables\n\nThe following expressions depend on the propagation time `t`.\n\n| variable | symbol | description |\n| :---------------------------- | :---------- | :------------------------------------------------------------------------------------------------------------------------ |\n| `t` | `t` | minutes elapsed since epoch (can be negative) |\n| `p22` | `p₂₂` | right ascension of the ascending node with neither Earth gravity resonance nor Sun and Moon contributions |\n| `p23` | `p₂₃` | argument of perigee with neither high altitude drag effects, Earth gravity resonance nor Sun and Moon contributions |\n| `orbit.inclination` | `I` | inclination at epoch plus `t` without the short-period effects of Earth gravity |\n| `orbit.right_ascension` | `Ω` | right ascension of the ascending node at epoch plus `t` without the short-period effects of Earth gravity |\n| `orbit.eccentricity` | `e` | eccentricity at epoch plus `t` without the short-period effects of Earth gravity |\n| `orbit.argument_of_perigee` | `ω` | argument of perigee at epoch plus `t` without the short-period effects of Earth gravity |\n| `orbit.mean_anomaly` | `M` | mean anomaly at epoch plus `t` without the short-period effects of Earth gravity |\n| `orbit.mean_motion` | `n` | mean motion at epoch plus `t` without the short-period effects of Earth gravity |\n| `a` | `a` | semi-major axis |\n| `p32` | `p₃₂` | partial expression of `aᵧₙ` |\n| `p33` | `p₃₃` | partial expression of `rₖ`, `ṙₖ` and `rḟₖ` |\n| `p34` | `p₃₄` | partial expression of `uₖ` |\n| `p35` | `p₃₅` | partial expression of the initial Kepler variable `p₃₈` |\n| `p36` | `p₃₆` | partial expression of `rₖ` and `rḟₖ` |\n| `p37` | `p₃₇` | partial expression of `aᵧₙ` and the initial Kepler variable `p₃₈` |\n| `axn` | `aₓₙ` | normalized linear eccentricity projected on the line of nodes |\n| `ayn` | `aᵧₙ` | normalized linear eccentricity projected on the normal to the line of nodes |\n| `p38` | `p₃₈` | initial Kepler variable (`U` in [[2]](#2), renamed to avoid confusion with the true anomaly plus argument of perigee `u`) |\n| `ew` | `(E + ω)ᵢ` | Kepler variable used in an iterative process to estimate the eccentric anomaly `E` |\n| `delta ` | `Δ(E + ω)ᵢ` | correction to the Kepler variable at iteration `i` |\n| `p39` | `p₃₉` | eccentricity at epoch plus `t` |\n| `pl` | `pₗ` | semi-latus rectum |\n| `p40` | `p₄₀` | normalized linear eccentricity projected on the semi-minor axis |\n| `r` | `r` | radius (distance to the focus) without the short-period effects of Earth gravity |\n| `r_dot` | `ṙ` | radius time derivative without the short-period effects of Earth gravity |\n| `b` | `β` | semi-minor axis over semi-major axis |\n| `p41` | `p₄₁` | partial expression of `p₄₂` and `p₄₃` |\n| `p42` | `p₄₂` | sine of `u` |\n| `p43` | `p₄₃` | cosine of `u` |\n| `u` | `u` | true anomaly plus argument of perigee without the short-period effects of Earth gravity |\n| `p44` | `p₄₄` | `sin(2 u)`, partial expression of `uₖ`, `Ωₖ` and `ṙₖ` |\n| `p45` | `p₄₅` | `cos(2 u)`, partial expression of `rₖ`, `Iₖ` and `rḟₖ` |\n| `p46` | `p₄₆` | partial expression of `rₖ`, `uₖ`, `Iₖ` and `Ωₖ` |\n| `rk` | `rₖ` | radius (distance to the focus) |\n| `uk` | `uₖ` | true anomaly plus argument of perigee |\n| `inclination_k` | `Iₖ` | inclination at epoch plus `t` |\n| `right_ascension_k` | `Ωₖ` | right ascension at epoch plus `t` |\n| `rk_dot` | `ṙₖ` | radius time derivative |\n| `rfk_dot` | `rḟₖ` | radius times the true anomaly derivative |\n| `u0` | `u₀` | x component of the position unit vector |\n| `u1` | `u₁` | y component of the position unit vector |\n| `u2` | `u₂` | z component of the position unit vector |\n| `prediction.position[0]` | `r₀` | x component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `prediction.position[1]` | `r₁` | y component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `prediction.position[2]` | `r₂` | z component of the position vector in km (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `prediction.velocity[0]` | `ṙ₀` | x component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `prediction.velocity[1]` | `ṙ₁` | y component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `prediction.velocity[2]` | `ṙ₂` | z component of the velocity vector in km.s⁻¹ (True Equator, Mean Equinox (TEME) of epoch reference frame) |\n| `p24` | `p₂₄` | mean anomaly without drag contributions in near earth propagation |\n| `p25` | `p₂₅` | partial expression of `ω` and `M` in near earth propagation |\n| `p26` | `p₂₆` | mean anomaly with elliptic correction and without drag contributions in near earth propagation |\n| `p27` | `p₂₇` | non-clamped eccentricity in near earth propagation |\n| `p28` | `p₂₈` | semi-major axis with resonance correction in deep space propagation |\n| `p29` | `p₂₉` | mean anomaly with resonance correction in deep space propagation |\n| `p31` | `p₃₁` | non-clamped eccentricity in deep space propagation |\n| `sidereal_time` | `θ` | sidereal time at epoch plus `t` |\n| `delta_t` | `Δt` | time step used in the integration of resonance effects of Earth gravity in min (either `720` or `-720`) |\n| `lambda_dot` | `λ̇ᵢ` | resonance effects of Earth gravity variable's time derivative at epoch plus `i Δt` |\n| `ni_dot` | `ṅᵢ` | mean motion time derivative at epoch plus `i Δt` |\n| `ni_ddot` | `n̈ᵢ` | mean motion second time derivative at epoch plus `i Δt` |\n| `ResonanceState::t` | `tᵢ` | resonance effects of Earth gravity integrator time (`i Δt`) |\n| `ResonanceState::mean_motion` | `nᵢ` | mean motion time derivative at epoch plus `Δt i` |\n| `ResonanceState::lambda` | `λᵢ` | resonance effects of Earth gravity variable at epoch plus `i Δt` |\n| `p30` | `p₃₀` | non-normalised `Ω` in Lyddane deep space propagation |\n\n#### Third-body initialization variables\n\nThe contribution of the Sun and the Moon to the orbital elements are calculated with a unique set of expressions. _src/third_body.rs_ provides a generic implementation of these expressions. Variables specific to the third body (either the Sun or the Moon) are annotated with `x`. In every other file, these variables are annotated with `s` if they correspond to solar perturbations, and `l` if they correspond to lunar perturbations.\n\nThe `aₓₙ`, `Xₓₙ`, `Zₓₙ` (`n ∈ ℕ`), `Fₓ₂` and `Fₓ₃` variables correspond to the `aₙ`, `Xₙ`, `Zₙ`, `F₂` and `F₃` variables in [[2]](#2). The added `x` highlights the dependence on the perturbing third body.\n\nThe following variables depend solely on epoch elements.\n\n| variable | symbol | description |\n| :-------------------------------------- | :------------- | :----------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n| `third_body_inclination_sine` | `sin Iₓ` | sine of the inclination of the Sun (`sin Iₛ`) or the Moon (`sin Iₗ`) |\n| `third_body_inclination_cosine` | `cos Iₓ` | cosine of the inclination of the Sun (`cos Iₛ`) or the Moon (`cos Iₗ`) |\n| `delta_right_ascension_sine` | `sin(Ω₀ - Ωₓ)` | sine of the difference between the right ascension of the ascending node of the satellite at epoch and the Sun's (`sin(Ω₀ - Ωₛ)`) or the Moon's (`sin(Ω₀ - Ωₗ)`) |\n| `delta_right_ascension_cosine` | `cos(Ω₀ - Ωₓ)` | cosine of the difference between the right ascension of the ascending node of the satellite at epoch and the Sun's (`cos(Ω₀ - Ωₛ)`) or the Moon's (`cos(Ω₀ - Ωₗ)`) |\n| `third_body_argument_of_perigee_sine` | `sin ωₓ` | sine of the argument of perigee of the Sun (`sin ωₛ`) or the Moon (`sin ωₗ`) |\n| `third_body_argument_of_perigee_cosine` | `cos ωₓ` | cosine of the argument of perigee of the Sun (`sin ωₛ`) or the Moon (`cos ωₗ`) |\n| `third_body_mean_anomaly_0` | `Mₓ₀` | mean anomaly at epoch of the Sun (`Mₛ₀`) or the Moon (`Mₗ₀`) |\n| `ax1` | `aₓ₁` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `ax3` | `aₓ₃` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `ax7` | `aₓ₇` | partial expression of multiple `aₓ₂` and `aₓ₅` |\n| `ax8` | `aₓ₈` | partial expression of multiple `aₓ₂` and `aₓ₅` |\n| `ax9` | `aₓ₉` | partial expression of multiple `aₓ₄` and `aₓ₆` |\n| `ax10` | `aₓ₁₀` | partial expression of multiple `aₓ₄` and `aₓ₆` |\n| `ax2` | `aₓ₂` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `ax4` | `aₓ₄` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `ax5` | `aₓ₅` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `ax6` | `aₓ₆` | partial expression of multiple `Xₓₙ` and `Zₓₙ` expressions |\n| `xx1` | `Xₓ₁` | partial expression of multiple `Zₓₙ` expressions, `kₓ₀`, `kₓ₁` and `ėₓ` |\n| `xx2` | `Xₓ₂` | partial expression of multiple `Zₓₙ` expressions, `kₓ₀`, `kₓ₁` and `ėₓ` |\n| `xx3` | `Xₓ₃` | partial expression of multiple `Zₓₙ` expressions, `kₓ₀`, `kₓ₁` and `ėₓ` |\n| `xx4` | `Xₓ₄` | partial expression of multiple `Zₓₙ` expressions, `kₓ₀`, `kₓ₁` and `ėₓ` |\n| `xx5` | `Xₓ₅` | partial expression of multiple `Zₓₙ` expressions |\n| `xx6` | `Xₓ₆` | partial expression of multiple `Zₓₙ` expressions |\n| `xx7` | `Xₓ₇` | partial expression of multiple `Zₓₙ` expressions |\n| `xx8` | `Xₓ₈` | partial expression of multiple `Zₓₙ` expressions |\n| `zx31` | `Zₓ₃₁` | partial expression of `Zₓ₃`, `kₓ₈` and `ω̇ₓ` |\n| `zx32` | `Zₓ₃₂` | partial expression of `Zₓ₂`, `kₓ₇` and `ω̇ₓ` |\n| `zx33` | `Zₓ₃₃` | partial expression of `Zₓ₃`, `kₓ₈` and `ω̇ₓ` |\n| `zx11` | `Zₓ₁₁` | partial expression of `kₓ₃` and `İₓ` |\n| `zx13` | `Zₓ₁₃` | partial expression of `kₓ₃` and `İₓ` |\n| `zx21` | `Zₓ₂₁` | partial expression of `kₓ₁₁` and `Ω̇ₓ` |\n| `zx23` | `Zₓ₂₃` | partial expression of `kₓ₁₁` and `Ω̇ₓ` |\n| `zx1` | `Zₓ₁` | partial expression of `kₓ₅` and `Ṁₓ` |\n| `zx3` | `Zₓ₃` | partial expression of `kₓ₅` and `Ṁₓ` |\n| `px0` | `pₓ₀` | partial expression of multiple `kₓₙ` expressions and `Ṁₓ` |\n| `px1` | `pₓ₁` | partial expression of multiple `kₓₙ` expressions and `İₓ` |\n| `px2` | `pₓ₂` | partial expression of multiple `kₓₙ` expressions and `ω̇ₓ` |\n| `px3` | `pₓ₃` | partial expression of multiple `kₓₙ` expressions and `ėₓ` |\n| `kx0` | `kₓ₀` | `Fₓ₂` coefficient of `δeₓ` |\n| `kx1` | `kₓ₁` | `Fₓ₃` coefficient of `δeₓ` |\n| `kx2` | `kₓ₂` | `Fₓ₂` coefficient of `δIₓ` |\n| `kx3` | `kₓ₃` | `Fₓ₃` coefficient of `δIₓ` |\n| `kx4` | `kₓ₄` | `Fₓ₂` coefficient of `δMₓ` |\n| `kx5` | `kₓ₅` | `Fₓ₃` coefficient of `δMₓ` |\n| `kx6` | `kₓ₆` | `sin fₓ` coefficient of `δMₓ` |\n| `kx7` | `kₓ₇` | `Fₓ₂` coefficient of `pₓ₄` |\n| `kx8` | `kₓ₈` | `Fₓ₃` coefficient of `pₓ₄` |\n| `kx9` | `kₓ₉` | `sin fₓ` coefficient of `pₓ₄` |\n| `kx10` | `kₓ₁₀` | `Fₓ₂` coefficient of `pₓ₅` |\n| `kx11` | `kₓ₁₁` | `Fₓ₃` coefficient of `pₓ₅` |\n| `third_body_dots.inclination` | `İₓ` | secular contribution of the Sun (`İₛ`) or the Moon (`İₗ`) to the inclination |\n| `third_body_right_ascension_dot` | `Ω̇ₓ` | secular contribution of the Sun (`Ω̇ₛ`) or the Moon (`Ω̇ₗ`) to the right ascension of the ascending node |\n| `third_body_dots.eccentricity` | `ėₓ` | secular contribution of the Sun (`ėₛ`) or the Moon (`ėₗ`) to the eccentricity |\n| `third_body_dots.agument_of_perigee` | `ω̇ₓ` | secular contribution of the Sun (`ω̇ₛ`) or the Moon (`ω̇ₗ`) to the argument of perigee |\n| `third_body_dots.mean_anomaly` | `Ṁₓ` | secular contribution of the Sun (`Ṁₛ`) or the Moon (`Ṁₗ`) to the mean anomaly |\n\n#### Third-body propagation variables\n\nThe following variables depend on the propagation time `t`.\n\n| variable | symbol | description |\n| :------------------------------ | :----- | :-------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n| `third_body_mean_anomaly` | `Mₓ` | mean anomaly of the Sun (`Mₛ`) or the Moon (`Mₗ`) |\n| `fx` | `fₓ` | third body true anomaly |\n| `fx2` | `Fₓ₂` | partial expression of the third body long-period periodic contribution |\n| `fx3` | `Fₓ₃` | partial expression of the third body long-period periodic contribution |\n| `third_body_delta_eccentricity` | `δeₓ` | long-period periodic contribution of the Sun (`δeₛ`) or the Moon (`δeₗ`) to the eccentricity |\n| `third_body_delta_inclination` | `δIₓ` | long-period periodic contribution of the Sun (`δIₛ`) or the Moon (`δIₗ`) to the inclination |\n| `third_body_delta_mean_mootion` | `δMₓ` | long-period periodic contribution of the Sun (`δMₛ`) or the Moon (`δMₗ`) to the mean motion |\n| `px4` | `pₓ₄` | partial expression of the long-period periodic contribution of the Sun (`pₛ₄`) or the Moon (`pₗ₄`) to the right ascension of the ascending node and the argument of perigee |\n| `px5` | `pₓ₅` | partial expression of the long-period periodic contribution of the Sun (`pₛ₅`) or the Moon (`pₗ₅`) to the right ascension of the ascending node |\n\n### Mathematical expressions\n\n#### UT1 to Julian conversion\n\nThe epoch (Julian years since UTC 1 January 2000 12h00) can be calculated with either the AFSPC formula:\n\n```\ny₂₀₀₀ = (367 yᵤ - ⌊7 (yᵤ + ⌊(mᵤ + 9) / 12⌋) / 4⌋ + 275 ⌊mᵤ / 9⌋ + dᵤ\n + 1721013.5\n + (((nsᵤ / 10⁹ + sᵤ) / 60 + minᵤ) / 60 + hᵤ) / 24\n - 2451545)\n / 365.25\n```\n\nor the more accurate version of the same formula:\n\n```\ny₂₀₀₀ = (367 yᵤₜ₁ - ⌊7 (yᵤₜ₁ + ⌊(mᵤₜ₁ + 9) / 12⌋) / 4⌋ + 275 ⌊mᵤₜ₁ / 9⌋ + dᵤₜ₁ - 730531) / 365.25\n + (3600 hᵤₜ₁ + 60 minᵤₜ₁ + sᵤₜ₁ - 43200) / (24 × 60 × 60 × 365.25)\n + nsᵤₜ₁ / (24 × 60 × 60 × 365.25 × 10⁹)\n```\n\n#### Common initialization\n\n```\na₁ = (kₑ / n₀)²ᐟ³\n\n 3 3 cos²I₀ - 1\n p₀ = - J₂ ---------------\n 4 (1 − e₀²)³ᐟ²\n\n𝛿₁ = p₀ / a₁²\n\n𝛿₀ = p₀ / (a₁ (1 - ¹/₃ 𝛿₁ - 𝛿₁² - ¹³⁴/₈₁ 𝛿₁³))²\n\nn₀\" = n₀ / (1 + 𝛿₀)\n\np₁ = cos I₀\n\np₂ = 1 − e₀²\n\nk₆ = 3 p₁² - 1\n\na₀\" = (kₑ / n₀\")²ᐟ³\n\np₃ = a₀\" (1 - e₀)\n\np₄ = aₑ (p₃ - 1)\n\np₅ = │ 20 if p₄ \u003c 98\n │ p₄ - 78 if 98 ≤ p₄ \u003c 156\n │ 78 otherwise\n\ns = p₅ / aₑ + 1\n\np₆ = ((120 - p₅) / aₑ)⁴\n\nξ = 1 / (a₀\" - s)\n\np₇ = p₆ ξ⁴\n\nη = a₀\" e₀ ξ\n\np₈ = |1 - η²|\n\np₉ = p₇ / p₈⁷ᐟ²\n\nC₁ = B* p₉ n₀\" (a₀\" (1 + ³/₂ η² + e₀ η (4 + η²))\n + ³/₈ J₂ ξ k₆ (8 + 3 η² (8 + η²)) / p₈)\n\np₁₀ = (a₀\" p₂)⁻²\n\nβ₀ = p₂¹ᐟ²\n\np₁₁ = ³/₂ J₂ p₁₀ n₀\"\n\np₁₂ = ¹/₂ p₁₁ J₂ p₁₀\n\np₁₃ = - ¹⁵/₃₂ J₄ p₁₀² n₀\"\n\np₁₄ = - p₁₁ p₁ + (¹/₂ p₁₂ (4 - 19 p₁²) + 2 p₁₃ (3 - 7 p₁²)) p₁\n\nk₁₄ = - ¹/₂ p₁₁ (1 - 5 p₁²) + ¹/₁₆ p₁₂ (7 - 114 p₁² + 395 p₁⁴)\n\np₁₅ = n₀\" + ¹/₂ p₁₁ β₀ k₆ + ¹/₁₆ p₁₂ β₀ (13 - 78 p₁² + 137 p₁⁴)\n\nC₄ = 2 B* n₀\" p₉ a₀\" p₂ (\n η (2 + ¹/₂ η²)\n + e₀ (¹/₂ + 2 η²)\n - J₂ ξ / (a p₈) (-3 k₆ (1 - 2 e₀ η + η² (³/₂ - ¹/₂ e₀ η))\n + ³/₄ (1 - p₁²) (2 η² - e₀ η (1 + η²)) cos 2 ω₀)\n\nk₀ = ⁷/₂ p₂ p₁₁ p₁ C₁\n\nk₁ = ³/₂ C₁\n\nΩ̇ = │ p₁₄ if n₀\" \u003e 2π / 225\n │ p₁₄ + (Ω̇ₛ + Ω̇ₗ) otherwise\n\nω̇ = │ k₁₄ if n₀\" \u003e 2π / 225\n │ k₁₄ + (ω̇ₛ + ω̇ₗ) otherwise\n\nṀ = │ p₁₅ if n₀\" \u003e 2π / 225\n │ p₁₅ + (Ṁₛ + Ṁₗ) otherwise\n```\n\n#### Near earth initialization\n\nDefined only if `n₀\" \u003e 2π / 225` (near earth).\n\n```\n 1 J₃\nk₂ = - - -- sin I₀\n 2 J₂\n\nk₃ = 1 - p₁²\n\nk₄ = 7 p₁² - 1\n\n │ 1 J₃ 3 + 5 p₁\nk₅ = │ - - -- sin I₀ -------- if |1 + p₁| \u003e 1.5 × 10⁻¹²\n │ 4 J₂ 1 + p₁\n │ 1 J₃ 3 + 5 p₁\n │ - - -- sin I₀ ----------- otherwise\n │ 4 J₂ 1.5 × 10⁻¹²\n```\n\n#### High altitude near earth initialization\n\nDefined only if `n₀\" \u003e 2π / 225` (near earth) and `p₃ ≥ 220 / (aₑ + 1)` (high altitude).\n\n```\nD₂ = 4 a₀\" ξ C₁²\n\np₁₆ = D₂ ξ C₁ / 3\n\nD₃ = (17 a + s) p₁₆\n\nD₄ = ¹/₂ p₁₆ a₀\" ξ (221 a₀\" + 31 s) C₁\n\nC₅ = 2 B* p₉ a₀\" p₂ (1 + 2.75 (η² + η e₀) + e₀ η³)\n\nk₁₁ = (1 + η cos M₀)³\n\nk₇ = sin M₀\n\nk₈ = D₂ + 2 C₁²\n\nk₉ = ¹/₄ (3 D₃ + C₁ (12 D₂ + 10 C₁²))\n\nk₁₀ = ¹/₅ (3 D₄ + 12 C₁ D₃ + 6 D₂² + 15 C₁² (2 D₂ + C₁²))\n```\n\n#### Elliptic high altitude near earth initialization\n\nDefined only if `n₀\" \u003e 2π / 225` (near earth), `p₃ ≥ 220 / (aₑ + 1)` (high altitude) and `e₀ \u003e 10⁻⁴` (elliptic).\n\n```\n J₃ p₇ ξ n₀\" sin I₀\nk₁₂ = - 2 B* cos ω₀ -- ----------------\n J₂ e₀\n\n 2 p₇ B*\nk₁₃ = - - -----\n 3 e₀ η\n```\n\n#### Deep space initialization\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space).\n\n```\ne₁₉₀₀ = 365.25 (t₀ + 100)\n\nsin Iₛ = 0.39785416\n\ncos Iₛ = 0.91744867\n\nsin(Ω₀ - Ωₛ) = sin Ω₀\n\ncos(Ω₀ - Ωₛ) = cos Ω₀\n\nsin ωₛ = -0.98088458\n\ncos ωₛ = 0.1945905\n\nMₛ₀ = (6.2565837 + 0.017201977 e₁₉₀₀) rem 2π\n\nΩₗₑ = 4.523602 - 9.2422029 × 10⁻⁴ e₁₉₀₀ rem 2π\n\ncos Iₗ = 0.91375164 - 0.03568096 Ωₗₑ\n\nsin Iₗ = (1 - cos²Iₗ)¹ᐟ²\n\nsin Ωₗ = 0.089683511 sin Ωₗₑ / sin Iₗ\n\ncos Ωₗ = (1 - sin²Ωₗ)¹ᐟ²\n\nωₗ = 5.8351514 + 0.001944368 e₁₉₀₀\n 0.39785416 sin Ωₗₑ / sin Iₗ\n + tan⁻¹ ------------------------------------------ - Ωₗₑ\n cos Ωₗ cos Ωₗₑ + 0.91744867 sin Ωₗ sin Ωₗₑ\n\nsin(Ω₀ - Ωₗ) = sin Ω₀ cos Ωₗ - cos Ω₀ sin Ωₗ\n\ncos(Ω₀ - Ωₗ) = cos Ωₗ cos Ω₀ + sin Ωₗ sin Ω₀\n\nMₗ₀ = (-1.1151842 + 0.228027132 e₁₉₀₀) rem 2π\n```\n\n#### Third body perturbations\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space).\n\nThe following variables are evaluated for two third bodies, the Sun (solar perturbations `s`) and the Moon (lunar perturbations `l`). Variables specific to the third body are annotated with `x`. In other sections, `x` is either `s` or `l`.\n\n```\naₓ₁ = cos ωₓ cos(Ω₀ - Ωₓ) + sin ωₓ cos Iₓ sin(Ω₀ - Ωₓ)\n\naₓ₃ = - sin ωₓ cos(Ω₀ - Ωₓ) + cos ωₓ cos Iₓ sin(Ω₀ - Ωₓ)\n\naₓ₇ = - cos ωₓ sin(Ω₀ - Ωₓ) + sin ωₓ cos Iₓ cos(Ω₀ - Ωₓ)\n\naₓ₈ = sin ωₓ sin Iₓ\n\naₓ₉ = sin ωₓ sin(Ω₀ - Ωₓ) + cos ωₓ cos Iₓ cos(Ω₀ - Ωₓ)\n\naₓ₁₀ = cos ωₓ sin Iₓ\n\naₓ₂ = aₓ₇ cos i₀ + aₓ₈ sin I₀\n\naₓ₄ = aₓ₉ cos i₀ + aₓ₁₀ sin I₀\n\naₓ₅ = - aₓ₇ sin I₀ + aₓ₈ cos I₀\n\naₓ₆ = - aₓ₉ sin I₀ + aₓ₁₀ cos I₀\n\nXₓ₁ = aₓ₁ cos ω₀ + aₓ₂ sin ω₀\n\nXₓ₂ = aₓ₃ cos ω₀ + aₓ₄ sin ω₀\n\nXₓ₃ = - aₓ₁ sin ω₀ + aₓ₂ cos ω₀\n\nXₓ₄ = - aₓ₃ sin ω₀ + aₓ₄ cos ω₀\n\nXₓ₅ = aₓ₅ sin ω₀\n\nXₓ₆ = aₓ₆ sin ω₀\n\nXₓ₇ = aₓ₅ cos ω₀\n\nXₓ₈ = aₓ₆ cos ω₀\n\nZₓ₃₁ = 12 Xₓ₁² - 3 Xₓ₃²\n\nZₓ₃₂ = 24 Xₓ₁ Xₓ₂ - 6 Xₓ₃ Xₓ₄\n\nZₓ₃₃ = 12 Xₓ₂² - 3 Xₓ₄²\n\nZₓ₁₁ = - 6 aₓ₁ aₓ₅ + e₀² (- 24 Xₓ₁ Xₓ₇ - 6 Xₓ₃ Xₓ₅)\n\nZₓ₁₃ = - 6 aₓ₃ aₓ₆ + e₀² (-24 Xₓ₂ Xₓ₈ - 6 Xₓ₄ Xₓ₆)\n\nZₓ₂₁ = 6 aₓ₂ aₓ₅ + e₀² (24.0 Xₓ₁ Xₓ₅ - 6 Xₓ₃ Xₓ₇)\n\nZₓ₂₃ = 6 aₓ₄ aₓ₆ + e₀² (24 Xₓ₂ Xₓ₆ - 6 Xₓ₄ Xₓ₈)\n\nZₓ₁ = 2 (3 (aₓ₁² + aₓ₂²) + Zₓ₃₁ e₀²) + p₁ Zₓ₃₁\n\nZₓ₃ = 2 (3 (aₓ₃² + aₓ₄²) + Zₓ₃₃ e₀²) + p₁ Zₓ₃₃\n\npₓ₀ = Cₓ / n₀\"\n\n 1 pₓ₀\npₓ₁ = - - ---\n 2 β₀\n\npₓ₂ = pₓ₀ β₀\n\npₓ₃ = - 15 e₀ pₓ₂\n\nΩ̇ₓ = │ 0 if I₀ \u003c 5.2359877 × 10⁻²\n │ or I₀ \u003e π - 5.2359877 × 10⁻²\n │ - nₓ pₓ₁ (Zₓ₂₁ + Zₓ₂₃) / sin I₀ otherwise\n\nkₓ₀ = 2 pₓ₃ (Xₓ₂ Xₓ₃ + Xₓ₁ Xₓ₄)\n\nkₓ₁ = 2 pₓ₃ (Xₓ₂ Xₓ₄ - Xₓ₁ Xₓ₃)\n\nkₓ₂ = 2 pₓ₁ (- 6 (aₓ₁ aₓ₆ + aₓ₃ aₓ₅) + e₀² (- 24 (Xₓ₂ Xₓ₇ + Xₓ₁ Xₓ₈) - 6 (Xₓ₃ Xₓ₆ + Xₓ₄ Xₓ₅)))\n\nkₓ₃ = 2 pₓ₁ (Zₓ₁₃ - Zₓ₁₁)\n\nkₓ₄ = - 2 pₓ₀ (2 (6 (aₓ₁ aₓ₃ + aₓ₂ aₓ₄) + Zₓ₃₂ e₀²) + p₁ Zₓ₃₂)\n\nkₓ₅ = - 2 pₓ₀ (Zₓ₃ - Zₓ₁)\n\nkₓ₆ = - 2 pₓ₀ (- 21 - 9 e₀²) eₓ\n\nkₓ₇ = 2 pₓ₂ Zₓ₃₂\n\nkₓ₈ = 2 pₓ₂ (Zₓ₃₃ - Zₓ₃₁)\n\nkₓ₉ = - 18 pₓ₂ eₓ\n\nkₓ₁₀ = - 2 pₓ₁ (6 (aₓ₄ aₓ₅ + aₓ₂ aₓ₆) + e₀² (24 (Xₓ₂ Xₓ₅ + Xₓ₁ Xₓ₆) - 6 (Xₓ₄ Xₓ₇ + Xₓ₃ Xₓ₈)))\n\nkₓ₁₁ = - 2 pₓ₁ (Zₓ₂₃ - Zₓ₂₁)\n\nİₓ = pₓ₁ nₓ (Zₓ₁₁ + Zₓ₁₃)\n\nėₓ = pₓ₃ nₓ (Xₓ₁ Xₓ₃ + Xₓ₂ Xₓ₄)\n\nω̇ₓ = pₓ₂ nₓ (Zₓ₃₁ + Zₓ₃₃ - 6) - cos I₀ Ω̇ₓ\n\nṀₓ = - nₓ pₓ₀ (Zₓ₁ + Zₓ₃ - 14 - 6 e₀²)\n```\n\n#### Resonant deep space initialization\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space) and either:\n\n- `0.0034906585 \u003c n₀\" \u003c 0.0052359877` (geosynchronous)\n- `8.26 × 10⁻³ ≤ n₀\" ≤ 9.24 × 10⁻³` and `e₀ ≥ 0.5` (Molniya)\n\nThe sidereal time `θ₀` at epoch can be calculated with either the IAU formula:\n\n```\nc₂₀₀₀ = y₂₀₀₀ / 100\n\nθ₀ = ¹/₂₄₀ (π / 180) (- 6.2 × 10⁻⁶ c₂₀₀₀³ + 0.093104 c₂₀₀₀²\n + (876600 × 3600 + 8640184.812866) c₂₀₀₀ + 67310.54841) mod 2π\n```\n\nor the AFSPC formula:\n\n```\nd₁₉₇₀ = 365.25 (y₂₀₀₀ + 30) + 1\n\nθ₀ = 1.7321343856509374 + 1.72027916940703639 × 10⁻² ⌊d₁₉₇₀ + 10⁻⁸⌋\n + (1.72027916940703639 × 10⁻² + 2π) (d₁₉₇₀ - ⌊d₁₉₇₀ + 10⁻⁸⌋)\n + 5.07551419432269442 × 10⁻¹⁵ d₁₉₇₀² mod 2π\n```\n\n```\nλ₀ = │ M₀ + Ω₀ + ω₀ − θ₀ rem 2π if geosynchronous\n │ M₀ + 2 Ω₀ − 2 θ₀ rem 2π otherwise\n\nλ̇₀ = │ p₁₅ + (k₁₄ + p₁₄) − θ̇ + (Ṁₛ + Ṁₗ) + (ω̇ₛ + ω̇ₗ) + (Ω̇ₛ + Ω̇ₗ) - n₀\" if geosynchronous\n │ p₁₅ + (Ṁₛ + Ṁₗ) + 2 (p₁₄ + (Ω̇ₛ + Ω̇ₗ) - θ̇) - n₀\" otherwise\n```\n\n#### Geosynchronous deep space initialization\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space) and `0.0034906585 \u003c n₀\" \u003c 0.0052359877` (geosynchronous orbit).\n\n```\np₁₇ = 3 (n / a₀\")²\n\n𝛿ᵣ₁ = p₁₇ (¹⁵/₁₆ sin²I₀ (1 + 3 p₁) - ³/₄ (1 + p₁))\n (1 + 2 e₀²) 2.1460748 × 10⁻⁶ / a₀\"²\n\n𝛿ᵣ₂ = 2 p₁₇ (³/₄ (1 + p₁)²)\n (1 + e₀² (- ⁵/₂ + ¹³/₁₆ e₀²)) 1.7891679 × 10⁻⁶\n\n𝛿ᵣ₃ = 3 p₁₇ (¹⁵/₈ (1 + p₁)³) (1 + e₀² (- 6 + 6.60937 e₀²))\n 2.2123015 × 10⁻⁷ / a₀\"²\n```\n\n#### Molniya deep space initialization\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space) and `8.26 × 10⁻³ ≤ n₀\" ≤ 9.24 × 10⁻³` and `e₀ ≥ 0.5` (Molniya).\n\n```\np₁₈ = 3 n₀\"² / a₀\"²\n\np₁₉ = p₁₈ / a₀\"\n\np₂₀ = p₁₉ / a₀\"\n\np₂₁ = p₂₀ / a₀\"\n\nF₂₂₀ = ³/₄ (1 + 2 p₁ + p₁²)\n\nG₂₁₁ = │ 3.616 - 13.247 e₀ + 16.29 e₀² if e₀ ≤ 0.65\n │ - 72.099 + 331.819 e₀ - 508.738 e₀² + 266.724 e₀³ otherwise\n\nG₃₁₀ = │ - 19.302 + 117.39 e₀ - 228.419 e₀² + 156.591 e₀³ if e₀ ≤ 0.65\n │ - 346.844 + 1582.851 e₀ - 2415.925 e₀² + 1246.113 e₀³ otherwise\n\nG₃₂₂ = │ - 18.9068 + 109.7927 e₀ - 214.6334 e₀² + 146.5816 e₀³ if e₀ ≤ 0.65\n │ - 342.585 + 1554.908 e₀ - 2366.899 e₀² + 1215.972 e₀³ otherwise\n\nG₄₁₀ = │ - 41.122 + 242.694 e₀ - 471.094 e₀² + 313.953 e₀³ if e₀ ≤ 0.65\n │ - 1052.797 + 4758.686 e₀ - 7193.992 e₀² + 3651.957 e₀³ otherwise\n\nG₄₂₂ = │ - 146.407 + 841.88 e₀ - 1629.014 e₀² + 1083.435 e₀³ if e₀ ≤ 0.65\n │ - 3581.69 + 16178.11 e₀ - 24462.77 e₀² + 12422.52 e₀³ otherwise\n\nG₅₂₀ = │ - 532.114 + 3017.977 e₀ - 5740.032 e₀² + 3708.276 e₀³ if e₀ ≤ 0.65\n │ 1464.74 - 4664.75 e₀ + 3763.64 e₀² if 0.65 \u003c e₀ \u003c 0.715\n │ - 5149.66 + 29936.92 e₀ - 54087.36 e₀² + 31324.56 e₀³ otherwise\n\nG₅₃₂ = │ - 853.666 + 4690.25 e₀ - 8624.77 e₀² + 5341.4 e₀³ if e₀ \u003c 0.7\n │ - 40023.88 + 170470.89 e₀ - 242699.48 e₀² + 115605.82 e₀³ otherwise\n\nG₅₂₁ = │ - 822.71072 + 4568.6173 e₀ - 8491.4146 e₀² + 5337.524 e₀³ if e₀ \u003c 0.7\n │ - 51752.104 + 218913.95 e₀ - 309468.16 e₀² + 146349.42 e₀³ otherwise\n\nG₅₃₃ = │ - 919.2277 + 4988.61 e₀ - 9064.77 e₀² + 5542.21 e₀³ if e₀ \u003c 0.7\n │ - 37995.78 + 161616.52 e₀ - 229838.2 e₀² + 109377.94 e₀³ otherwise\n\nD₂₂₀₋₁ = p₁₈ 1.7891679 × 10⁻⁶ F₂₂₀ (- 0.306 - 0.44 (e₀ - 0.64))\n\nD₂₂₁₁ = p₁₈ 1.7891679 × 10⁻⁶ (³/₂ sin²I₀) G₂₁₁\n\nD₃₂₁₀ = p₁₉ 3.7393792 × 10⁻⁷ (¹⁵/₈ sin I₀ (1 - 2 p₁ - 3 p₁²)) G₃₁₀\n\nD₃₂₂₂ = p₁₉ 3.7393792 × 10⁻⁷ (- ¹⁵/₈ sin I₀ (1 + 2 p₁ - 3 p₁²)) G₃₂₂\n\nD₄₄₁₀ = 2 p₂₀ 7.3636953 × 10⁻⁹ (35 sin²I₀ F₂₂₀) G₄₁₀\n\nD₄₄₂₂ = 2 p₂₀ 7.3636953 × 10⁻⁹ (³¹⁵/₈ sin⁴I₀) G₄₂₂\n\nD₅₂₂₀ = p₂₁ 1.1428639 × 10⁻⁷ (³¹⁵/₃₂ sin I₀\n (sin²I₀ (1 - 2 p₁ - 5 p₁²)\n + 0.33333333 (- 2 + 4 p₁ + 6 p₁²))) G₅₂₀\n\nD₅₂₃₂ = p₂₁ 1.1428639 × 10⁻⁷ (sin I₀\n (4.92187512 sin²I₀ (- 2 - 4 p₁ + 10 p₁²)\n + 6.56250012 (1 + p₁ - 3 p₁²))) G₅₃₂\n\nD₅₄₂₁ = 2 p₂₁ 2.1765803 × 10⁻⁹ (⁹⁴⁵/₃₂ sin I₀\n (2 - 8 p₁ + p₁² (- 12 + 8 p₁ + 10 p₁²))) G₅₂₁\n\nD₅₄₃₃ = 2 p₂₁ 2.1765803 × 10⁻⁹ (⁹⁴⁵/₃₂ sin I₀\n (- 2 - 8 p₁ + p₁² (12 + 8 p₁ - 10 p₁²))) G₅₃₃\n```\n\n#### Common propagation\n\nThe following values depend on the propagation time `t` (minutes since epoch).\n\nNamed conditions have the following meaning:\n\n- `near earth`: `n₀\" ≤ 2π / 225`\n- `low altitude near earth`: `near earth` and `p₃ \u003c 220 / (aₑ + 1)`\n- `high altitude near earth`: `near earth` and `p₃ ≥ 220 / (aₑ + 1)`\n- `elliptic high altitude near earth`: `high altitude near earth` and `e₀ \u003e 10⁻⁴`\n- `non-elliptic near earth`: `low altitude near earth` or `high altitude near earth` and `e₀ ≤ 10⁻⁴`\n- `deep space`: `n₀\" \u003e 2π / 225`\n- `non-Lyddane deep space`: `deep space` and `I ≥ 0.2`\n- `Lyddane deep space`: `deep space` and `I \u003c 0.2`\n- `AFSPC Lyddane deep space`: `Lyddane deep space` and use the same expression as the original AFSPC implementation, with an `ω` discontinuity at `p₂₂ = 0`\n\n```\np₂₂ = Ω₀ + Ω̇ t + k₀ t²\n\np₂₃ = ω₀ + ω̇ t\n\nI = │ I₀ if near earth\n │ I₀ + İ t + (δIₛ + δIₗ) otherwise\n\nΩ = │ p₂₂ if near earth\n │ p₂₂ + (pₛ₅ + pₗ₅) / sin I if non-Lyddane deep space\n │ p₃₀ + 2π if Lyddane deep space and p₃₀ + π \u003c p₂₂ rem 2π\n │ p₃₀ - 2π if Lyddane deep space and p₃₀ - π \u003e p₂₂ rem 2π\n │ p₃₀ otherwise\n\ne = │ 10⁻⁶ if near earth and p₂₇ \u003c 10⁻⁶\n │ p₂₇ if near earth and p₂₇ ≥ 10⁻⁶\n │ 10⁻⁶ + (δeₛ + δeₗ) if deep space and p₃₁ \u003c 10⁻⁶\n │ p₃₁ + (δeₛ + δeₗ) otherwise\n\nω = │ p₂₃ - p₂₅ if elliptic high altitude near earth\n │ p₂₃ if non-elliptic near earth\n │ p₂₃ + (pₛ₄ + pₗ₄) - cos I (pₛ₅ + pₗ₅) / sin I if non-Lyddane deep space\n │ p₂₃ + (pₛ₄ + pₗ₄) + cos I ((p₂₂ rem 2π) - Ω)\n │ - (δIₛ + δIₗ) (p₂₂ mod 2π) sin I if AFSPC Lyddane deep space\n │ p₂₃ + (pₛ₄ + pₗ₄) + cos I ((p₂₂ rem 2π) - Ω)\n │ - (δIₛ + δIₗ) (p₂₂ rem 2π) sin I otherwise\n\nM = │ p₂₆ + n₀\" (k₁ t² + k₈ t³ + t⁴ (k₉ + t k₁₀) if high altitude near earth\n │ p₂₄ + n₀\" k₁ t² if low altitude near earth\n │ p₂₉ + (δMₛ + δMₗ) + n₀\" k₁ t² otherwise\n\n\na = │ a₀\" (1 - C₁ t - D₂ t² - D₃ t³ - D₄ t⁴)² if high altitude near earth\n │ a₀\" (1 - C₁ t)² if low altitude near earth\n │ p₂₈ (1 - C₁ t)² otherwise\n\nn = kₑ / a³ᐟ²\n\np₃₂ = │ k₂ if near earth\n │ 1 J₃\n │ - - -- sin I othewise\n │ 2 J₂\n\np₃₃ = │ k₃ if near earth\n │ 1 - cos²I othewise\n\np₃₄ = │ k₄ if near earth\n │ 7 cos²I - 1 otherwise\n\np₃₅ = │ k₅ if near earth\n │ 1 J₃ 3 + 5 cos I\n │ - - -- sin I ----------- if deep space and |1 + cos I| \u003e 1.5 × 10⁻¹²\n │ 4 J₂ 1 + cos I\n │ 1 J₃ 3 + 5 cos I\n │ - - -- sin I ----------- otherwise\n │ 4 J₂ 1.5 × 10⁻¹²\n\np₃₆ = │ k₆ if near earth\n │ 3 cos²I - 1 otherwise\n\np₃₇ = 1 / (a (1 - e²))\n\naₓₙ = e cos ω\n\naᵧₙ = e sin ω + p₃₇ p₃₂\n\np₃₈ = M + ω + p₃₇ p₃₅ aₓₙ rem 2π\n\n(E + ω)₀ = p₃₈\n\n p₃₈ - aᵧₙ cos (E + ω)ᵢ + aₓₙ sin (E + ω)ᵢ - (E + ω)ᵢ\nΔ(E + ω)ᵢ = ---------------------------------------------------\n 1 - cos (E + ω)ᵢ aₓₙ - sin (E + ω)ᵢ aᵧₙ\n\n(E + ω)ᵢ₊₁ = (E + ω)ᵢ + Δ(E + ω)ᵢ|[-0.95, 0.95]\n\nE + ω = │ (E + ω)₁₀ if ∀ j ∈ [0, 9], Δ(E + ω)ⱼ ≥ 10⁻¹²\n │ (E + ω)ⱼ otherwise, with j the smallest integer | Δ(E + ω)ⱼ \u003c 10⁻¹²\n\np₃₉ = aₓₙ² + aᵧₙ²\n\npₗ = a (1 - p₃₉)\n\np₄₀ = aₓₙ sin(E + ω) - aᵧₙ cos(E + ω)\n\nr = a (1 - (aₓₙ cos(E + ω) + aᵧₙ sin(E + ω)))\n\nṙ = a¹ᐟ² p₄₀ / r\n\nβ = (1 - p₃₉)¹ᐟ²\n\np₄₁ = p₄₀ / (1 + β)\n\np₄₂ = a / r (sin(E + ω) - aᵧₙ - aₓₙ p₄₁)\n\np₄₃ = a / r (cos(E + ω) - aₓₙ + aᵧₙ p₄₁)\n\n p₄₂\nu = tan⁻¹ ---\n p₄₃\n\np₄₄ = 2 p₄₃ p₄₂\n\np₄₅ = 1 - 2 p₄₂²\n\np₄₆ = (¹/₂ J₂ / pₗ) / pₗ\n\nrₖ = r (1 - ³/₂ p₄₆ β p₃₆) + ¹/₂ (¹/₂ J₂ / pₗ) p₃₃ p₄₅\n\nuₖ = u - ¹/₄ p₄₆ p₃₄ p₄₄\n\nΩₖ = Ω + ³/₂ p₄₆ cos I p₄₄\n\nIₖ = I + ³/₂ p₄₆ cos I sin I p₄₅\n\nṙₖ = ṙ + n (¹/₂ J₂ / pₗ) p₃₃ / kₑ\n\nrḟₖ = pₗ¹ᐟ² / r + n (¹/₂ J₂ / pₗ) (p₃₃ p₄₅ + ³/₂ p₃₆) / kₑ\n\nu₀ = - sin Ωₖ cos Iₖ sin uₖ + cos Ωₖ cos uₖ\n\nu₁ = cos Ωₖ cos Iₖ sin uₖ + sin Ωₖ cos uₖ\n\nu₂ = sin Iₖ sin uₖ\n\nr₀ = rₖ u₀ aₑ\n\nr₁ = rₖ u₁ aₑ\n\nr₂ = rₖ u₂ aₑ\n\nṙ₀ = (ṙₖ u₀ + rḟₖ (- sin Ωₖ cos Iₖ cos uₖ - cos Ωₖ sin uₖ)) aₑ kₑ / 60\n\nṙ₁ = (ṙₖ u₁ + rḟₖ (cos Ωₖ cos Iₖ cos uₖ - sin Ωₖ sin uₖ)) aₑ kₑ / 60\n\nṙ₂ = (ṙₖ u₂ + rḟₖ (sin Iₖ cos uₖ)) aₑ kₑ / 60\n```\n\n#### Near earth propagation\n\nDefined only if `n₀\" \u003e 2π / 225` (near earth).\n\n```\np₂₄ = M₀ + Ṁ t\n\np₂₇ = | e₀ - (C₄ t + C₅ (sin p₂₆ - k₇)) if high altitude\n | e₀ - C₄ t otherwise\n```\n\n#### High altitude near earth propagation\n\nDefined only if `n₀\" \u003e 2π / 225` (near earth) and `p₃ ≥ 220 / (aₑ + 1)` (high altitude).\n\n`elliptic` means `e₀ \u003e 10⁻⁴`.\n\n```\np₂₅ = k₁₃ ((1 + η cos p₂₄)³ - k₁₁) + k₁₂ t\n\np₂₆ = │ p₂₄ + p₂₅ if elliptic\n │ p₂₄ otherwise\n```\n\n#### Deep space propagation\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space).\n\n```\np₂₈ = │ (kₑ / (nⱼ + ṅⱼ (t - tⱼ) + ¹/₂ n̈ⱼ (t - tⱼ)²))²ᐟ³ if geosynchronous or Molniya\n │ a₀\" otherwise\n\np₂₉ = │ λⱼ + λ̇ⱼ (t - tⱼ) + ¹/₂ ṅᵢ (t - tⱼ)² - p₂₂ - p₂₃ + θ if geosynchronous\n │ λⱼ + λ̇ⱼ (t - tⱼ) + ¹/₂ ṅᵢ (t - tⱼ)² - 2 p₂₂ + 2 θ if Molniya\n │ M₀ + Ṁ t otherwise\n\nj is │ the largest positive integer | tⱼ ≤ t if t \u003e 0\n │ the smallest negative integer | tⱼ ≥ t if t \u003c 0\n │ 0 otherwise\n\np₃₁ = e₀ + ė t - C₄ t\n```\n\n#### Third body propagation\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space).\n\nThe following variables are evaluated for two third bodies, the Sun (solar perturbations `s`) and the Moon (lunar perturbations `l`). Variables specific to the third body are annotated with `x`. In other sections, `x` is either `s` or `l`.\n\n```\nMₓ = Mₓ₀ + nₓ t\n\nfₓ = Mₓ + 2 eₓ sin Mₓ\n\nFₓ₂ = ¹/₂ sin²fₓ - ¹/₄\n\nFₓ₃ = - ¹/₂ sin fₓ cos fₓ\n\nδeₓ = kₓ₀ Fₓ₂ + kₓ₁ Fₓ₃\n\nδIₓ = kₓ₂ Fₓ₂ + kₓ₃ Fₓ₃\n\nδMₓ = kₓ₄ Fₓ₂ + kₓ₅ Fₓ₃ + kₓ₆ sin fₓ\n\npₓ₄ = kₓ₇ Fₓ₂ + kₓ₈ Fₓ₃ + kₓ₉ sin fₓ\n\npₓ₅ = kₓ₁₀ Fₓ₂ + kₓ₁₁ Fₓ₃\n```\n\n#### Resonant deep space propagation\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space) and either:\n\n- `0.0034906585 \u003c n₀\" \u003c 0.0052359877` (geosynchronous)\n- `8.26 × 10⁻³ ≤ n₀\" ≤ 9.24 × 10⁻³` and `e₀ ≥ 0.5` (Molniya)\n\n```\nθ = θ₀ + 4.37526908801129966 × 10⁻³ t rem 2π\n\nΔt = │ |Δt| if t \u003e 0\n │ -|Δt| if t \u003c 0\n │ 0 otherwise\n\nλ̇ᵢ = nᵢ + λ̇₀\n\nṅᵢ = │ 𝛿ᵣ₁ sin(λᵢ - λ₃₁) + 𝛿ᵣ₂ sin(2 (λᵢ - λ₂₂)) + 𝛿ᵣ₃ sin(3 (λᵢ - λ₃₃)) if geosynchronous\n │ Σ₍ₗₘₚₖ₎ Dₗₘₚₖ sin((l - 2 p) ωᵢ + m / 2 λᵢ - Gₗₘ) otherwise\n\nn̈ᵢ = │ (𝛿ᵣ₁ cos(λᵢ - λ₃₁) + 𝛿ᵣ₂ cos(2 (λᵢ - λ₂₂)) + 𝛿ᵣ₃ cos(3 (λᵢ - λ₃₃))) λ̇ᵢ if geosynchronous\n │ (Σ₍ₗₘₚₖ₎ m / 2 Dₗₘₚₖ cos((l - 2 p) ωᵢ + m / 2 λᵢ - Gₗₘ)) λ̇ᵢ otherwise\n\n(l, m, p, k) ∈ {(2, 2, 0, -1), (2, 2, 1, 1), (3, 2, 1, 0),\n (3, 2, 2, 2), (4, 4, 1, 0), (4, 4, 2, 2), (5, 2, 2, 0),\n (5, 2, 3, 2), (5, 4, 2, 1), (5, 4, 3, 3)}\n\ntᵢ₊₁ = tᵢ + Δt\n\nnᵢ₊₁ = nᵢ + ṅᵢ Δt + n̈ᵢ (Δt² / 2)\n\nλᵢ₊₁ = λᵢ + λ̇ᵢ Δt + ṅᵢ (Δt² / 2)\n```\n\n#### Lyddane deep space propagation\n\nDefined only if `n₀\" ≤ 2π / 225` (deep space) and `I \u003c 0.2` (Lyddane).\n\n```\n sin I sin p₂₂ + (pₛ₅ + pₗ₅) cos p₂₂ + (δIₛ + δIₗ) cos I sin p₂₂\np₃₀ = tan⁻¹ -------------------------------------------------------------\n sin I cos p₂₂ - (pₛ₅ + pₗ₅) sin p₂₂ + (δIₛ + δIₗ) cos I cos p₂₂\n```\n\n## References\n\n\u003ca id=\"1\"\u003e[1]\u003c/a\u003e David A. Vallado, Paul Crawford, R. S. Hujsak and T. S. Kelso, \"Revisiting Spacetrack Report #3\", presented at the AIAA/AAS Astrodynamics Specialist Conference, Keystone, CO, 2006 August 21–24, https://doi.org/10.2514/6.2006-6753\n\n\u003ca id=\"2\"\u003e[2]\u003c/a\u003e F. R. Hoots, P. W. Schumacher Jr. and R. A. Glover, \"History of Analytical Orbit Modeling in the U. S. Space Surveillance System\", Journal of Guidance, Control, and Dynamics, 27(2), 174–185, 2004, https://doi.org/10.2514/1.9161\n\n\u003ca id=\"3\"\u003e[3]\u003c/a\u003e F. R. Hoots and R. L. Roehrich, \"Spacetrack Report No. 3: Models for propagation of NORAD element sets\", U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, 1980, https://www.celestrak.com/NORAD/documentation/\n\n\u003ca id=\"4\"\u003e[4]\u003c/a\u003e R. S. Hujsak, \"A Restricted Four Body Solution for Resonating Satellites Without Drag\", Project SPACETRACK, Rept. 1, U.S. Air Force Aerospace Defense Command, Colorado Springs, CO, Nov. 1979, https://doi.org/10.21236/ada081263\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fneuromorphicsystems%2Fsgp4","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fneuromorphicsystems%2Fsgp4","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fneuromorphicsystems%2Fsgp4/lists"}